They don't mean upgrading to a suite at a "regular" hotel; there are chains specializing in renting out "home-style" rooms ("extended-stay" is the industry term, I believe), and I've seen them referred to suite hotels sometimes.
The biggest chains have these sub-brands in the space, based off three minutes of googling/wikiing:
I've never seen an extended-stay hotel in a neighborhood that I would visit for leisure. They're usually in pretty ugly, utilitarian spots - off a cloverleaf or behind a strip mall - serving business travelers.
That's probably true in the US; much less so in Asia or places with good public transit.
There's a Hyatt House I pass by sometimes right next to the Shibuya crossing; though I'm guessing that one definitely does not have the price advantage.
I don't know how you would easily book them if you're not in the system, but what you're describing is a timeshare rental - suite rooms in resort-like environments.
Right. Residence Inn is one of my goto brands. Not always certainly but I tend to be staying at mid-rangeish hotels anyway and Residence is usually in that band, especially outside of downtowns.
I have not found this to be the case at all. Getting a suite with a kitchen usually involves paying a large premium to a regular hotel room.